Question 1087209: Nielsen Rating: The television show nbc Sunday night football broadcast a game between the colts and patriots and received a share of 22, meaning that among the tv sets in use, 22% are turned to that game (based on data from Nielsen Media Research). An advertiser wants to obtain a second opinion by conducting its own surveys and a pilot survey begins with 20 households having tv sets in use at the time of that same nbc Sunday night football broadcast.
a. Find the probability that none of the households are turned to nbc Sunday night football (3 digits)
b. Find the probability that at least one of the households is turned to nbc Sunday night football.(3 digits)
c. Find the probability that at most one of the households is turned to nbc Sunday night football (3 digits)
d. If at most one household is turned to nbc Sunday night football, does it appear that the 22% share value is wrong? Why or why not?
Please show me how you got the answer. (I need help)
Answer by mathmate(429)   (Show Source):
You can put this solution on YOUR website! Questoin:
Nielsen Rating: The television show nbc Sunday night football broadcast a game between the colts and patriots and received a share of 22, meaning that among the tv sets in use, 22% are turned to that game (based on data from Nielsen Media Research). An advertiser wants to obtain a second opinion by conducting its own surveys and a pilot survey begins with 20 households having tv sets in use at the time of that same nbc Sunday night football broadcast.
a. Find the probability that none of the households are turned to nbc Sunday night football (3 digits)
b. Find the probability that at least one of the households is turned to nbc Sunday night football.(3 digits)
c. Find the probability that at most one of the households is turned to nbc Sunday night football (3 digits)
d. If at most one household is turned to nbc Sunday night football, does it appear that the 22% share value is wrong? Why or why not?
Please show me how you got the answer. (I need help)
Solution:
The situation satisfies requirements to solve using the binomial distribution, verified as follows:
1. Bernoulli trials, i.e. exactly two possible outcomes (success=tuned to program, else failure)
2. Number of trials is known before and constant throughout the experiment (20), i.e. independent of outcomes.
3. All trials are independent of each other (inferred from context)
4. Probability of success is known, and remain constant throughout trials (given 22% is the official value for the population).
Since all criteria are satisfied, we can model the given situation with binomial distribution, where the probability of x successes out of N trials each with probability of success p is given by
P(x)=C(N,x)(p^x)(1-p)^(N-x)
and,
C(N,x) is number of combinations of selecting x objects out of N.
p=0.22
n=20
(a) P(X=0)
=P(X=0)
=C(20,0)*(0.22^0)*(0.78^20)
=0.00695
(b) P(X>=1)
=1-P(X=0)
=0.99305
(c) P(X<=1)
=P(X=0)+P(X=1)
=0.00695+C(20,1)(0.22^1)(0.78^19)
=0.00695+0.03920
=0.04615
(d) if at most one household is tuned, then there is a reasonable doubt to the value 22% since 0.0465 is less than the normal acceptable range of "19 time out of 20". However, it is NOT impossible that 22% is still correct in view of the fact that n=20 is not a reasonable sample size.
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